| Title:
|
Approximation Spacesin Non-commutative Generalizations of $MV$-algebras (English) |
| Author:
|
RACHŮNEK, Jiří |
| Author:
|
ŠALOUNOVÁ, Dana |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
54 |
| Issue:
|
2 |
| Year:
|
2015 |
| Pages:
|
83-92 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Generalized MV-algebras (= GMV-algebras) are non-commutative generalizations of MV-algebras. They are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued fuzzy logic. The paper investigates approximation spaces in GMV-algebras based on their normal ideals. (English) |
| Keyword:
|
MV-algebra |
| Keyword:
|
GMV-algebra |
| Keyword:
|
rough set |
| Keyword:
|
approximation space |
| Keyword:
|
normal ideal |
| Keyword:
|
congruence |
| MSC:
|
06D35 |
| idZBL:
|
Zbl 1347.06014 |
| idMR:
|
MR3469693 |
| . |
| Date available:
|
2015-12-21T17:12:49Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144765 |
| . |
| Reference:
|
[1] Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bull. Polish Acad. Sci., Math. 42 (1994), 251–254. Zbl 0834.68102, MR 1811855 |
| Reference:
|
[2] Burris, S., Sankapanavar, H. P.: A Course in Universal Algebra. Springer-Verlag, New York, Heidelberg, Berlin, 1981. MR 0648287 |
| Reference:
|
[3] Cataneo, G., Ciucci, D.: Algebraic structures for rough sets. Lect. Notes Comput. Sci. 3135 (2004), 208–252. 10.1007/978-3-540-27778-1_12 |
| Reference:
|
[4] Cignoli, R. L. O., D’Ottaviano, I. M. L., Mundici, D.: Algebraic Foundation of Many-valued Reasoning. Kluwer Acad. Publ., Dordrecht, Boston, London, 2000. MR 1786097 |
| Reference:
|
[5] Ciucci, D.: On the axioms of residuated structures independence dependencies and rough approximations. Fundam. Inform. 69 (2006), 359–387. Zbl 1100.06011, MR 2350921 |
| Reference:
|
[6] Ciucci, D.: A unifying abstract approach for rough models. In: C., Wang, et al: RSKT 2008, 5009, Springer-Verlag, Heidelberg, 2008, 371–378. |
| Reference:
|
[7] Davvaz, B.: Roughness in rings. Inform. Sci. 164 (2004), 147–163. Zbl 1072.16042, MR 2076574, 10.1016/j.ins.2003.10.001 |
| Reference:
|
[8] Davvaz, B.: Roughness based on fuzzy ideals. Inform. Sci. 176 (2006), 2417–2437. Zbl 1112.03049, MR 2247407, 10.1016/j.ins.2005.10.001 |
| Reference:
|
[9] Estaji, A. A., Khodaii, S., Bahrami, S.: On rough set and fuzzy sublattice. Inform. Sci. 181 (2011), 3981–3994. Zbl 1242.06008, MR 2818650, 10.1016/j.ins.2011.04.043 |
| Reference:
|
[10] Georgescu, G., Iorgulescu, A.: Pseudo MV-algebras. Multi. Val. Logic 6 (2001), 95–135. Zbl 1014.06008, MR 1817439 |
| Reference:
|
[11] Kondo, M., M.: Algebraic approach to generalized rough sets. In: D., Slezak, et al: RSFDGrC 2005, LNAI, 5009, Springer-Verlag, Berlin, Heidelberg, 2005, 132–140. Zbl 1134.68509, MR 2167787 |
| Reference:
|
[12] Kuroki, N.: Rough ideals in semigroups. Inform. Sci. 100 (1997), 139–163. Zbl 0916.20046, MR 1446627, 10.1016/S0020-0255(96)00274-5 |
| Reference:
|
[13] Leoreanu-Fotea, V., Davvaz, B.: Roughness in n-ary hypergroups. Inform. Sci. 178 (2008), 4114–4124. Zbl 1187.20071, MR 2454652, 10.1016/j.ins.2008.06.019 |
| Reference:
|
[14] Leuştean, I.: Non-commutative Łukasiewicz propositional logic. Arch. Math. Logic 45 (2006), 191–213. Zbl 1096.03020, MR 2209743, 10.1007/s00153-005-0297-8 |
| Reference:
|
[15] Li, T. J., Leung, Y., Zhang, W. X.: Generalized fuzzy rough approximation operators based on fuzzy coverings. Int. J. Approx. Reason. 48 (2008), 836–856. Zbl 1186.68464, MR 2437954, 10.1016/j.ijar.2008.01.006 |
| Reference:
|
[16] Li, X., Liu, S.: Matroidal approaches to rough sets via closure operators. Inter. J. Approx. Reason. 53 (2012), 513–527. Zbl 1246.68233, MR 2903082 |
| Reference:
|
[17] Liu, G. L., Zhu, W.: The algebraic structures of generalized rough set theory. Inform. Sci. 178 (2008), 4105–4113. Zbl 1162.68667, MR 2454651, 10.1016/j.ins.2008.06.021 |
| Reference:
|
[18] Pawlak, Z.: Rough sets. Inter. J. Inf. Comput. Sci. 11 (1982), 341–356. Zbl 0525.04005, MR 0703291 |
| Reference:
|
[19] Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. System Theory, Knowledge Engineering and Problem Solving 9, Kluwer Academic Publ., Dordrecht, 1991. Zbl 0758.68054 |
| Reference:
|
[20] Pawlak, Z., Skowron, A.: Rudiments of rough sets. Inform. Sci. 177 (2007), 3–27. Zbl 1142.68549, MR 2272732, 10.1016/j.ins.2006.06.003 |
| Reference:
|
[21] Pawlak, Z., Skowron, A.: Rough sets: some extensions. Inform. Sci. 177 (2007), 28–40. Zbl 1142.68550, MR 2272733, 10.1016/j.ins.2006.06.006 |
| Reference:
|
[22] Pawlak, Z., Skowron, A.: Rough sets and Boolean reasoning. Inform. Sci. 177 (2007), 41–73. Zbl 1142.68551, MR 2272734, 10.1016/j.ins.2006.06.007 |
| Reference:
|
[23] Pei, D.: On definable concepts of rough set models. Inform. Sci. 177 (2007), 4230–4239. Zbl 1126.68076, MR 2346788, 10.1016/j.ins.2007.01.020 |
| Reference:
|
[24] Rachůnek, J.: A non-commutative generalization of MV-algebras. Czechoslovak Math. J. 52 (2002), 255–273. Zbl 1012.06012, MR 1905434, 10.1023/A:1021766309509 |
| Reference:
|
[25] Radzikowska, A. M., Kerre, E. E.: Fuzzy rough sets based on residuated lattices. In: Transactions on Rough Sets II, Lecture Notes in Computer Sciences, 3135, Springer-Verlag, Berlin, Heidelberg, 2004, 278–296. Zbl 1109.68118 |
| Reference:
|
[26] Rasouli, S., Davvaz, B.: Roughness in MV-algebras. Inform. Sci. 180 (2010), 737–747. Zbl 1194.06009, MR 2574551, 10.1016/j.ins.2009.11.008 |
| Reference:
|
[27] She, Y. H., Wang, G. J.: An approximatic approach of fuzzy rough sets based on residuated lattices. Comput. Math. Appl. 58 (2009), 189–201. MR 2535981, 10.1016/j.camwa.2009.03.100 |
| Reference:
|
[28] Xiao, Q. M., Zhang, Z. L.: Rough prime ideals and rough fuzzy prime ideals in semigroups. Inform. Sci. 176 (2006), 725–733. Zbl 1088.20041, MR 2201320, 10.1016/j.ins.2004.12.010 |
| Reference:
|
[29] Yang, L., Xu, L.: Algebraic aspects of generalized approximation spaces. Int. J. Approx. Reason. 51 (2009), 151–161. Zbl 1200.06004, MR 2565213, 10.1016/j.ijar.2009.10.001 |
| Reference:
|
[30] Zhu, P.: Covering rough sets based on neighborhoods: An approach without using neighborhoods. Int. J. Approx. Reason. 52 (2011), 461–472. Zbl 1229.03047, MR 2771972, 10.1016/j.ijar.2010.10.005 |
| Reference:
|
[31] Zhu, W.: Relationship between generalized rough set based on binary relation and covering. Inform. Sci. 179 (2009), 210–225. MR 2473013, 10.1016/j.ins.2008.09.015 |
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